Introduction to Evolutionary Algorithms Yong Wang Lecturer, Ph.D. School of Information Science and Engineering, Central South University

2  Evolutionary Algorithms  Genetic Algorithm  Evolutionary Strategy  Evolutionary Programming  Particle Swarm Optimization  Differential Evolution Outline of My Talk The main branches of evolutionary algorithms

3 Outline of My Talk  Evolutionary Algorithms  Genetic Algorithm  Evolutionary Strategy  Evolutionary Programming  Particle Swarm Optimization  Differential Evolution

4 Why Evolutionary Algorithms (EAs)? (1/2) -Search surface having multiple modals The optimal solution Is it the optimal solution? The optimal solution! -Search surface having single modal What is the difficult of the traditional optimization methods?

5 Why Evolutionary Algorithms (EAs)? (2/2) - How about searching from different directions?  It’s a basic idea of EAs The optimal solution!

6 ant colony algorithm particle swarm optimization genetic algorithm What Are Evolutionary Algorithms? Evolutionary algorithms are intelligent optimization and search techniques inspired by nature

7 The Framework of Evolutionary Algorithms Population Parent Set Selection the first individual the second individual the NPth individual New Solutions Crossover + Mutation Replacement x y f(x,y)

8 The Characteristics of Evolutionary Algorithms Search for the optimal solution from many points rather than one point Choose the individuals based on the fitness function and do not need the gradient information of the problems Use the random probability transition rule rather than the deterministic transition rule

9 The Applications of Evolutionary Algorithms EAs can be used to solve different kinds of optimization problems For example: unconstrained single-objective optimization problems constrained single-objective optimization problems unconstrained multi-objective optimization problems constrained multi-objective optimization problems EAs

10 Unconstrained Single-objective Optimization Problems min

11 Constrained Single-objective Optimization Problems

12 Unconstrained Multi-objective Optimization Problems min

13 Constrained Multi-objective Optimization Problems

14 The Top Journals in Evolutionary Computation Community IEEE Transactions on Evolutionary Computation (TEC, since 1997, 6 issues per year, about 60 papers per year) Evolutionary Computation (EC, since 1993, 4 issues per year, about 20 papers per year) IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics (SMCB)

15 The Top Conferences in Evolutionary Computation Community IEEE Congress on Evolutionary Computation (CEC, Frequency: every year) Genetic and Evolutionary Computation Conference (GECCO, Frequency: every year) Evolutionary Multi-Criterion Optimization (EMO, Frequency: every two years)

16 Outline of My Talk  Evolutionary Algorithms  Genetic Algorithm  Evolutionary Strategy  Evolutionary Programming  Particle Swarm Optimization  Differential Evolution

17 Introduction (1/2) What are genetic algorithms? –Proposed by Professor J. Holland in the 1960s –Take their inspiration from Darwin’s theory of evolution, i.e., natural selection and survival of the fittest in the biological world ( 物竞天择，适者生存 )

18 Introduction (2/2) The basic component of GAs –chromosome or string, which is also called an individual in a population string (chromosome) character, Feature, (gene ) feature value (allele) string position (locus) Schema Highly fit, short-defining-length (BB) Building Block Please note that the next six slides of GAs were provided by Dr. Chang Wook Ahn ( I just did some minor revisions.

19 The analogy between biological evolution and simple genetic algorithm Simple Genetic Algorithm (1/4) POPULATION MATING POOL NEW POPULATION OFFSPRING MATING (crossover) MATES SELECTED

20 NO Simple Genetic Algorithm (2/4) Flowchart of a simple genetic algorithm Define: Parameters Fitness function Create population Fitness evaluation GA operators STOP YES Convergence Test Selection + Crossover + Mutation

21 Simple Genetic Algorithm (3/4) Three operators of the genetic algorithms 1. Selection:  Individuals are copied according to their fitness function values  An artificial version of natural selection  Roulette wheel selection, Tournament selection, etc Roulette wheel selection

22 Simple Genetic Algorithm (4/4) 2. Crossover:  Members of the newly reproduced strings in the mating pool are mated at random  After choosing a cross site at random, the partial information of the two selected strings are exchanged 3. Mutation:  Changing a “1” to a”0” or visa versa, occurring points are randomly selected  Escaping from the converging into local optimal solutions

23 Example of Genetic Algorithm Consider the following Complete Graph K 4 (node 1  node 4) 2341234 13241324 134134 141 2341234 12341234 134134 1414 1414 12341234 134134 12341234 12341234 12341234 12341234 1234123 2341234 12341234 12341234 12341234 12341234 12341234 12341234 1234123 STOP Shortest Path selection crossover selection crossover

24 Real-coded Genetic Algorithms Algorithmic framework Crossover operator Steady-state genetic algorithms B Q C R Selection Plan Generation Plan Replacement Plan R’ Update Plan ①② ③ ④

25 Crossover Operators (1/4) The offspring generated by unimodal normal distribution crossover (UNDX)

26 Crossover Operators (2/4) The offspring generated by simplex crossover (SPX)

27 Crossover Operators (3/4) The offspring generated by parent-centric recombination (PCX)

28 Crossover Operators (4/4) The main characteristics of the crossover operators –UNDX and SPX are mean-centric recombination –PCX is parent-centric recombination –These three crossover operators are multi-parent crossover, i.e., unlike the common crossover operators, all of them need more than two parents to take part in crossover. These three crossover operators are the most competitive crossover operators in real-coded genetic algorithms.

29 Outline of My Talk  Evolutionary Algorithms  Genetic Algorithm  Evolutionary Strategy  Evolutionary Programming  Particle Swarm Optimization  Differential Evolution

30 Evolutionary Strategy (ES) ES was proposed by I. Rechenberg in 1964 There are several versions of ES, for instance –( )-ES offspring µ parents µ parents and offspring µ parents

31 The Framework of ( )-ES Algorithmic framework k=1,…, objective function value

32 Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) In addition to ( )-ES and ( )-ES, two advanced versions of ES have been proposed by N. Hansen –CMA-ES –Restart CMA-ES (R-CMA-ES) Currently, CMA-ES is the most well-known ES and R- CMA-ES is the most competitive ES

33 The Main Idea of CMA-ES

34 Outline of My Talk  Evolutionary Algorithms  Genetic Algorithms  Evolutionary Strategy  Evolutionary Programming  Particle Swarm Optimization  Differential Evolution

35 Introduction to the EP EP was proposed by L. J. Fogel in 1966 objective function value Algorithmic framework objective function value

36 Cauchy Mutation VS Gaussian Mutation X. Yao, Y. Liu and G. Lin, ``Evolutionary programming made faster,'' IEEE Transactions on Evolutionary Computation, vol. 3, no. 2, pp , Gaussian mutation Cauchy mutation

37 Cauchy Distribution VS Gaussian Distribution amplitude tail

38 Theoretical Background (1/3)

39 Theoretical Background (2/3) Mean value theorem of integrals

40 Theoretical Background (3/3) absolute value Remark: the similar analysis can be carried out for Cauchy distribution

41 Outline of My Talk  Evolutionary Algorithms  Genetic Algorithm  Evolutionary Strategy  Evolutionary Programming  Particle Swarm Optimization  Differential Evolution

42 Particle Swarm Optimization (PSO) was invented by James Kennedy and Russ Eberhart in 1995 PSO takes inspiration of the motion of a flock of birds PSO has been used to solve many kinds of problems In PSO, each potential solution is regarded as a particle. Particle Swarm Optimization J. Kennedy R. Eberhart

43 The Movement Equations of the Particles w denotes the inertia weight, c 1 and c 2 are the acceleration constants, r 1 and r 2 are two separately generated uniformly distributed random numbers in the range [0,1]. denotes the j th variable of the i th particle at generation t. x i,j t pbest i,j t gbest i,j t vi,jtvi,jt v i,j t+1 x i,j t+1 the personal best the best of the swarm the movement of each variable

44 Analysis of PSO Let W. Hu, Z. LI. A simpler and more effective particle swarm optimization algorithm, Journal of Software, 2007,18(4):

45 Swarm Topology Since the particles interact with each other, the swarm topology is very important for the performance of PSO R. Mendes, J. Kennedy, and J. Neves. The fully informed particle swarm: simpler, maybe better. IEEE Transactions on Evolutionary Computation, vol. 8, no. 3, pp , 2004.

46 Outline of My Talk  Evolutionary Algorithms  Genetic Algorithm  Evolutionary Strategy  Evolutionary Programming  Particle Swarm Optimization  Differential Evolution

47 Differential Evolution (1/2) Differential evolution (DE), proposed by Storn and Price in 1995, is one of the main branches of evolutionary algorithms (EAs). In principle, DE is a kind of real-coded greedy genetic algorithm (GA). DE includes three main operators, i.e., mutation operator, crossover operator, and selection operator.

48 Differential Evolution (2/2) The algorithmic framework of DE Remark: mutation + crossover = trial vector generation strategy the target vector

49 The Mutation Operators rand/1 rand/2 best/1 best/2 current-to-best/1 current-to-rand/1 Remark: r1, r2, r3, r4, and r5 are different indexes uniformly randomly selected from, is the best individuals in the current population, and F is the scaling factor. the base vector the difference vector the fashion the base vector has been selected the number of the difference vector

50 The Characteristics of the Mutation Operators (1/3) rand/1 Characteristics –“rand/1” is the most commonly used mutation operator in the literature. –All vectors for mutation are selected from the population at random and, consequently, it has no bias to any special search directions and chooses new search directions in a random manner. –It usually demonstrates slow convergence speed and bears stronger exploration capability.

51 The Characteristics of the Mutation Operators (2/3) rand/2 Characteristics –In “rand/2”, two difference vectors are added to the base vector, which might lead to better perturbation than the strategies with only one difference vector. –It can generate more different trial vectors than the “rand/1” mutation operator. –When using “rand/2”, the diversity of the population can be kept, however, it has a side effect on the convergence speed of DE.

52 The Characteristics of the Mutation Operators (3/3) best/1 best/2 current-to-best/1 Characteristics –“best/1”, “best/2” and “current-to-best/1”, usually have the fast convergence speed and perform well when solving unimodal problems. –They are easier to get stuck at a local optimum and thereby lead to a premature convergence when solving multimodal problems. –The “best/1” is a degenerated case of the“current-to-best/1” with the first scaling factor F being equal to 1.

53 The Crossover Operators (1/2) Binomial crossover

54 The Crossover Operators (2/2) Exponential crossover

55 The Characteristics of the Crossover Operators Characteristics –Binomial crossover is similar to discrete crossover in GA. –Exponential crossover is functionally equivalent to two-point crossover in GA. –Exponential crossover has the capability in maintaining the linkage among variables and the building block. –Binomial crossover may destroy building block.

56 DE Variations By combining different mutation operators and different crossover operators, we can obtain different DE variants. DE/x/y/z –DE: differential evolution –x: the fashion the base vector has been selected –y: the number of the difference vector –z: the type of the crossover operator; “bin” represents the binomial crossover and “exp” represents the exponential crossover DE/rand/1/bin, DE/rand/1/exp, DE/rand/2/bin, …

57 The illustrative graph of DE/rand/1/bin the triangle denotes the trial vector

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